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Mirrors > Home > MPE Home > Th. List > zlmval | Structured version Visualization version GIF version |
Description: Augment an abelian group with vector space operations to turn it into a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) |
Ref | Expression |
---|---|
zlmval.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmval.m | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
zlmval | ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmval.w | . 2 ⊢ 𝑊 = (ℤMod‘𝐺) | |
2 | elex 3212 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
3 | oveq1 6657 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet 〈(Scalar‘ndx), ℤring〉) = (𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) | |
4 | fveq2 6191 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺)) | |
5 | zlmval.m | . . . . . . 7 ⊢ · = (.g‘𝐺) | |
6 | 4, 5 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · ) |
7 | 6 | opeq2d 4409 | . . . . 5 ⊢ (𝑔 = 𝐺 → 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉 = 〈( ·𝑠 ‘ndx), · 〉) |
8 | 3, 7 | oveq12d 6668 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉) = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
9 | df-zlm 19853 | . . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉)) | |
10 | ovex 6678 | . . . 4 ⊢ ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉) ∈ V | |
11 | 8, 9, 10 | fvmpt 6282 | . . 3 ⊢ (𝐺 ∈ V → (ℤMod‘𝐺) = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
12 | 2, 11 | syl 17 | . 2 ⊢ (𝐺 ∈ 𝑉 → (ℤMod‘𝐺) = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
13 | 1, 12 | syl5eq 2668 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 Scalarcsca 15944 ·𝑠 cvsca 15945 .gcmg 17540 ℤringzring 19818 ℤModczlm 19849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-zlm 19853 |
This theorem is referenced by: zlmlem 19865 zlmsca 19869 zlmvsca 19870 zlmds 30008 zlmtset 30009 |
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